Fractional order-pid-controller design

Computers and Mathematics with Applications ( ) –
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Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Fractional-order PIλ
Dµ
controller design
Reyad El-Khazali∗
ECE Department, Khalifa University of Science, Technology, and Research, Sharjah, United Arab Emirates
a r t i c l e i n f o
Keywords:
Fractional-order systems
PIλ
Dµ
controller
Fractional-order lead-lag controller
a b s t r a c t
This paper introduces a new design method of fractional-order proportional–derivative
(FOPD) and fractional-order proportional–integral–derivative (FOPID) controllers. A
biquadratic approximation of a fractional-order differential operator is used to introduce
a new structure of finite-order FOPID controllers. Using the new FOPD controllers, the
controlled systems can achieve the desired phase margins without migrating the gain
crossover frequency of the uncontrolled system. This may not be guaranteed when using
FOPID controllers. The proposed FOPID controller has a smaller number of parameters to
tune than its existing counterparts. A systematic design procedure is identified in terms
of the desired phase and the gain margins of the controlled systems. The viability of the
design methods is verified using a simple numerical example.
© 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Fractional calculus is a generalization of the classical integer-order calculus that includes integro-differential operators of
fractional orders. Fractional-order calculus has gained extensive attention lately since many systems in chemistry, physics,
and in engineering manifest a memory effect and they are best described by fractional-order dynamics [1–3]. Due to the
increase in system applications, considerable attention has been given to exact and numerical solutions of fractional-order
integro-differential equations [4]. The existence and uniqueness of solutions of linear and nonlinear fractional-order integro-
differential equations were discussed by Babakhani and Baleanu [5].
The performance of fractional-order systems can be manipulated by implementing integer or fractional-order control
algorithms. In many applications, it has been demonstrated that fractional-order controllers have superseded their integer-
order counterparts [6–8].
The proportional–integral–derivative (PID) controller is one such controller that has been successfully used in industrial
applications for several decades. The popularity of the PID controller lies in the simplicity of the design procedures and in
the effectiveness of its system performance [9]. A fractional-order PID controller (FOPID), on the other hand, was introduced
in [10,3,11]; it is a generalization of the conventional integer-order PID controller. It is denoted by PIλ
Dµ
, where λ and µ are
two additional parameters to the integral and the derivative components of the conventional PID controller, thus increasing
the complexity of tuning these parameters.
Several attempts to find an optimum setting for the five different parameters of the fractional PIλ
Dµ
controller, in order to
achieve predefined design requirements, are presented in [7,12]. The tuning rules of Ziegler–Nichols for an FOPID controller
were reported in [13]. New tuning algorithms for FOPID controllers are recently presented in [6,14]. The validity of an
optimum FOPID controller tuned by a particle swarm was also demonstrated by Karimi et al. [15] to control the automatic
voltage regulator (AVR) of a power system.
The existing design and tuning algorithms of FOPID controllers were demonstrated via numerical simulations using
MATLAB/SIMULINK toolboxes [16]; however, hardware realization of FOPID controllers is more challenging [8,17,18]. The
proposed method intends to look for new realizable forms and new tuning rules for FOPID controllers.
∗ Tel.: +971 6 597 8822.
E-mail addresses: khazali@kustar.ac.ae, khazali1957@gmail.com.
0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.camwa.2013.02.015

2 R. El-Khazali / Computers and Mathematics with Applications ( ) –
The rest of the paper is organized as follows. Section 2 describes a fundamental overview of fractional-order systems.
Section 3 provides a new fractional-order lead controller design, which is a rudimental to a new FOPID controller. Section 4
introduces a new design method of FOPID controllers. The main ideas of the design are described by numerical examples
embedded when necessary. Finally, Section 5 summarizes the main points of this work.
2. Fractional-order systems overview
Fractional-order systems, which are based on fractional-order calculus [19,20], are a generalization of dynamical systems
that exhibit non-Newtonian behavior [21]. The integer-order dynamics describe special and smaller class of fractional-order
systems. Consequently, fractional-order controllers, as demonstrated by many researchers, such as Podlubny et al. [11] and
El-Khazali et al. [22], outperformed their integer-order counterparts.
Fractional-order systems are described by an n-term non-homogeneous fractional-order differential equation (FDE) of
the form
anDαn
y(t) + an−1Dαn−1
y(t) + · · · a1Dα1
y(t) + a0Dα0
y(t) = bmDβm
u(t) + b1Dβ1
u(t) + b0Dβ0
u(t), (1)
where Dα
= 0Dα
t is the Caputo fractional derivative of order αk; k = 1, 2, . . . , n, and where βl (l = 0, 1, 2, . . . , m) are
arbitrary constants.
One may assume, without loss of generality, that 0 = α0 < α1 < α2 · · · < αn, and 0 = β0 < β1 < β2 · · · < βm. The
Laplace transform of the fractional-order derivative, Dα
y(t); 0 < α ≤ 1, is given by Podlubny [3]:
 ∞
0
Dα
y(t)e−st
dt = sα
Y(s) −
N
k=0
sα−k−1
y(k)
(0), (2)
while the fractional-order integral of y is denoted by 0I−α
t y(t), and the sum in the right-hand side in (2) is omitted [12].
Therefore, for zero initial conditions, the transfer function of the fractional-order system described in (1) is given by
Gp(s) =
bmsβm + bm−1sβm−1 + · · · + b1sβ1 + b0sβ0
ansαn + an−1sαn−1 + · · · a1sα1 + a0sα0
. (3)
The goal of this work is to design an FOPID controller of the form
Gc (s) = Kp +
Ki
sλ
+ Kdsµ
. (4)
The complexity of this controller is evident due to the increase in the number of control parameters. There are five
different parameters (Kp, Ki, Kd, λ, and µ) that have to be tuned, which increases the flexibility of achieving preset design
requirements such as steady-state errors, phase and gain margins, and robustness.
The challenge of this work is to develop a realizable FOPID controller that exhibits a robust performance with a smaller
number of parameters, yet achieving the same design requirements. The key point is to look for acceptable and realizable
approximations to the differential operators, sλ
, and sµ
. The next section introduces a new design technique of a fractional-
order PDµ
controller that paves the way for a new design technique of an FOPID controller.
3. Fractional-order lead controller design
The proportional–derivative controller represents a class of a typical lead controller. Lead compensators are usually
cascaded with uncompensated plants to add a leading phase to stabilize and to reshape the plant’s frequency response.
The conventional lead compensator adds additional phase and gain to the uncontrolled system by carefully selecting its
poles and zeros in the complex plane [23]. Usually, the design process does not succeed at the first attempt and it requires
a trial and error process to achieve the design requirements.
To alleviate the problem of selecting the poles and zeros of PDµ
controllers, a biquadratic structure of the fractional-
order differential operator can be used to approximate the performance of PDµ
controllers within an operating frequency
band [18]. Once the first biquadratic structure is designed, higher-order PDµ
controllers are obtained by cascading several
modules. Thus, it can be considered as a modular controller design, in which several modules (each of biquadratic transfer
function) are automatically cascaded and normalized at different cut-off frequencies to improve the robustness of the
controlled system by widening the flatness of its phase response.
Now, consider the approximation of a fractional operator, sµ
, by the following biquadratic approximation [24]:
sµ
≈
a0s2
+ a1s + a2
a2s2 + a1s + a0
≡ T(s) ≡
N(s)
D(s)
, (5)
where a0, a1, and a2 are real constants.
Selecting a0, a1, and a2 properly can approximate a fractional differential (integral) operator within a band-limited
frequency spectrum, i.e., it can be used to design a PDµ
(or a proportional–integral PIµ
fractional-order) controller [18].

R. El-Khazali / Computers and Mathematics with Applications ( ) – 3
a b
Fig. 1. Bode diagram of T(s) for (a) a0 < a2, and (b) a0 > a2.
In order to change the operating points of the control system, one can shift the crossover frequency of (5) from ω =
1 rad/s to any desired frequency at ωc ; i.e., T(s/ωc ) = a0(s/ωc )2+a1(s/ωc )+a2
a2(s/ωc )2+a1(s/ωc )+a0
. Clearly, |T(ω/ωc )| = 1, and its phase contribution
at ω = ωc is equal to
ϕT(s) = tan−1
(a1/(a2 − a0)) − tan−1
(a1/(a0 − a2)). (6)
Obviously, when a0 > a2, Eq. (6) implies that ϕT(s) = tan−1
(a1/(a2 − a0)) − tan−1
(a1/(a0 − a2)) = π − 2 tan−1
(a1/(a0 −
a2)) > 0. Therefore, the biquadratic form in (5) exhibits a differentiation action around ωc . Conversely, when a2 > a0,
then ϕT(s) = tan−1
(a1/(a2 − a0)) − tan−1
(a1/(a0 − a2)) = −π + 2 tan−1
(a1/(a0 − a2)) < 0, and that approximates a
fractional-order integrator around its corner frequency.
To further verify the aforementioned arguments, Fig. 1(b) shows the Bode diagram of the biquadratic form T(s) =
2.707s2 +4.828s+0.707
0.707s2+4.828s+2.707
, where a0 = 2.707 > a2 = 0.707, while Fig. 1(a), on the other hand, shows the Bode diagram of its
reciprocal; i.e., T(s) = 0.707s2 +4.828s+2.707
2.707s2+4.828s+0.707
, where a0 = 0.707 < a2 = 2.707. Clearly, both approximations exhibit differential
(integral) action within a frequency band centered at ωc . A simple test on the pole-zero map of T(s) for both cases show that
the poles and zeros alternate. The zeros lead the poles when a0 > a2, while the zeros lag the poles when a0 < a2.
One may use the continuous fraction expansion (CFE) method introduced in [25] to replace the constants a0, a1, and a2,
by the following values [24]:



a0 = µ2
+ 3µ + 2
a2 = µ2
− 3µ + 2
a1 = β(1 − µ2
) + 6,
(7)
which results in a biquadratic approximation of uncontrollable phase and magnitude errors. The parameter β in (7) is
introduced to tune the quadratic approximation given by (5).
Substituting (7) into (5) yields a biquadratic transfer function that depends on β and the fractional order µ, i.e.,
Gc (s, µ) =
N(s, µ)
D(s, µ)
=
(µ2
+ 3µ + 2)s2
+ {β(1 − µ2
) + 6}s + (µ2
− 3µ + 2)
(µ2 − 3µ + 2)s2 + {β(1 − µ2) + 6}s + (µ2 + 3µ + 2)
. (8)
Eq. (8) exhibits an almost equiripple phase behavior around ωc . To improve this approximation, one may choose the
constants a0, a1, and a2 that can flatten the phase angle of (8) as follows [18]:
a0 = k1(1 + µ) + k2(µµ
+ µ)
a2 = k1(1 − µ) + k2(µµ
− µ),
(9)
and a1 is selected to obtain an exact phase at ωc [18]:
a1 = (a2 − a0) tan((2 + µ)π/4); 0 < µ < 1. (10)
The real constants k1 and k2 are chosen to widen the operating points of the controllers. Notice that, when µ = 1, then
a0 = 2(k1 + k2), a2 = 0, a1 = 2(k1 + k2), and T(s) = s, as expected.

Fig. 2. Frequency response approximation of s0.5
using (8) and (11) for k1 = k2 = 0.1.
Substituting from (9) and (10) into (5) yields a new form of a biquadratic approximation to sµ
that depends on k1, k2 and
µ, i.e.,
Gc (s, µ) =
a0s2
− {2µ(k1 + k2) tan(2 + µ)π/4}s + a2
a2s2 − {2µ(k1 + k2) tan(2 + µ)π/4}s + a0
. (11)
The tuning parameters k1 and k2 in (11) allow one to achieve a flat frequency response which cannot be obtained by using
(8). Using (9) and (10), the leading (or lagging) phase contribution of (11) at ω = ωc is exactly equal to

N(s/ωc )
D(s/ωc )

= ±µπ/2. (12)
Fig. 2 shows the frequency response of (8) and (11) for µ = 0.5 and k1 = k2 = 0.1. Obviously, the approximation in
(11) yields a straight gain and a flat phase compared to its counterpart in (8), but over a narrower bandwidth. Selecting the
optimum values of k1 and k2 that widen the bandwidth is left for future consideration.
Fig. 2 justifies the use of the biquadratic approximation in (11), which mimics a PDµ
controller. Cascading (11) by a gain
Kc adds another parameter to improve the steady-state performance of the controlled system; i.e., the transfer function of
the PDµ
may be chosen as
Gc (s/ωc , µ) = Kc
N(s/ωc , µ)
D(s/ωc , µ)
= Kc
a0(s/ωc )2
+ {(a2 − a0) tan(2 + µ)π/4}(s/ωc ) + a2
a2(s/ωc )2 + {(a2 − a0) tan(2 + µ)π/4}(s/ωc ) + a0
. (13)
Hence, at ω = ωc , Eq. (13) yields



Gc

jωc
ωc
, µ



 = Kc , Gc

jωc
ωc
, µ

=
µπ
2
. (14)
3.1. PDµ
controller design
The FOPID controller exhibits an integro-differential action within a desired operating spectrum. Designing a PDµ
controller constitutes an initial step toward achieving this goal using the form in (13). Let Gp(jω) be the frequency response
of a typical plant. Let θp and gp be the phase angle and the gain margin of the uncontrolled system at the open-loop gain-
crossover frequency, ωcg , and the phase-crossover frequency, ωcp, respectively. Using (13), the characteristic equation of the
controlled system is described by T(jω) = 1 + Gc (jω, µ)Gp(jω) = 0, and the desired specifications of the controlled system
can be limited to the following.
(a) Achieving a desired phase margin (∅m) at (ωcg ); i.e.,
Arg

Gc

jω
ωcg
, µ

Gp

jω
ωcg

=
µπ
2
+ θp = −π + ∅m . (15)

(b) Maintaining a desired gain margin, gm, of the closed-loop system at the phase crossover frequency, (ωcp), of the open-
loop system; thus,
1


Gc

jω
ωcp
, µ

Gp(jω/ωcp)



= gm. (16)
Using (15) and (16), the order and the gain of the PDµ
controller are, respectively, given by
µ =
−π + ∅m −θp
π/2
, Kc =
1
gmgp
. (17)
Remark 1. To obtain a minimum phase biquadratic approximation using (9), (10) and (13), and to have a stable closed-
loop system, one must use a controller with a fractional order less than unity. In this case, a single stage controller will be
used. However, when µ ≥ 1, one must cascade several compensators, each of fractional order less than one, according to
following rules:
Gc

s
ωc
, µ

=
N(s, µ)
D(s, µ)
; 0 < µ < 1 (18)
Gc

s
ωc
, µ

=
Nm+1
(s, µ/(m + 1))
Dm+1(s, µ/(m + 1))
; m < µ < m + 1, m > 1. (19)
Example 1. Consider an open-loop system described by Gp(s) = 0.25
s3+s2 . Suppose it is desired to design a PDµ
controller to
stabilize the uncontrolled system, and to obtain a phase margin ∅m = 60° and a gain margin of gm ≥ 10 dB.
The phase angle of the open-loop system at its crossover frequency, ωcg = 0.475 rad/s, is equal to θp = −205.4°. Clearly, the
system is unstable. Using (17), the required order of the PDµ
compensator is equal to µ = 0.9489. Hence, a single controller
of order µ = 0.9489 normalized at ωcg = 0.475 rad/s will be used. Choosing k1 = 0.25, k2 = 0.5, and Kc = 1, Eq. (13)
yields the following desired controller:
Gc (s) =
1.438s2
+ 0.7329s + 0.003161
0.014s2 + 0.7329s + 0.3246
.
Fig. 3 shows the Bode diagram of the controlled system. Clearly, its phase margin is exactly equal to 60° at ωcg = 0.475 rad/s,
and the gain margin is 40 dB > 10 dB. The design requirements are then met using a single module of a PD0.9489
controller.
The designed controller is characterized by a realizable second-order biquadratic transfer function [18]. Observe that the
main feature of the proposed PDµ
algorithm is that the crossover frequency of the controlled system coincides with that of
the open-loop system.
Fig. 3. The Bode diagrams of both the open-loop and the controlled systems.

4. PIλ
Dµ
controller design
A realizable integer-order PID controller can be described by the following transfer function [23]:
GI (s) = Kp +
1
Tis
+
Tds
s + τ
, (20)
where Kp, Ti, Td, and τ must be selected and then tuned to meet the design requirements.
Similarly, the transfer function of an FOPID controller is given by Podlubny [11]:
GF (s) = Kp +
1
Tisλ
+ Tdsµ
. (21)
Since a PDµ
controller is a band-limited one, substituting from (13) into (21) yields a finite-order realizable FOPID controller.
In order to simplify the design procedure, the FOPID controller in (21) is simplified by assuming that µ = λ and Ti = Td.
Remark 2. The order of the integral part in (21) should not be constrained to add an additional pole at the origin, since this
limits the order of the integral part to (λ + 1), which affects the low-frequency response of the controlled system. Since the
poles and zeros of the biquadratic forms never reside at the origin, the chances to pull the poles of the controlled system
toward the left is higher than when having a pole at the origin, Moreover, if one requires a controller of fractional order
greater than 1, a cascaded biquadratic structure will be used in this case according to (19).
To benefit from the form in (5), Eq. (21) is slightly modified to take the following structure:
GF (s) = PIµ
Dµ
= Kp

2 +
1
Tisµ + Tisµ

, (22)
which can be rewritten as
GF (s) =
Kp(1 + Tisµ
)2
Tisµ
= Kc
(1 + Tisµ
)2
sµ
; Kc =
Kp
Ti
. (23)
Obviously, there are three parameters (Kc , Ti, and µ) in (23) that must be carefully selected to meet the design requirements.
The effectiveness of the design algorithm becomes evident when substituting (5) into (23) to yield the following finite-order
FOPID realizable controller:
GF (s) = Kc
(D(s) + TiN(s))2
N(s)D(s)
. (24)
Notice that, as Ti → 0, GF (s) → Kc
D(s)
N(s)
, which describes a realizable band-limited fractional-order integrator of order
µ. Conversely, as Ti → ∞, GF (s) → Kc
N(s)
D(s)
, and that represents a band-limited fractional-order differentiator; thus, (24)
combines a integro-differential action.
Substituting from (5) into (24) yields
GF (s) =
Kc ((a2 + Tia0)s2
+ a1(1 + Ti)s + (a0 + Tia2))2
(a0s2 + a1s + a2)(a2s2 + a1s + a0)
. (25)
Let ϕp be the phase margin of the uncontrolled plant, and let ϕc = Arg(GF (s)) be the phase contribution of the controller
in (25); then, for a desired phase margin of the controlled system, ∅m, the required controller phase angle, ϕc , is calculated
from
ϕc = ∅m −ϕp − π. (26)
Using (12), the phase contribution of (25) at ωcg is
Ti =



tan
ϕc
2

+ tan

2+µ
π/4

tan
ϕc
2

− tan

2+µ
π/4




; ϕc ̸=
(2 + µ)
π/8
. (27)
Equivalently, for a specific integrator time constant, the phase angle, ϕc , can also be found from
ϕc = 2 tan−1



(Ti + 1) tan

2+µ
π/4

Ti − 1



; Ti ̸= 1. (28)

Fig. 4. Frequency response of the controlled system using a PI0.7778
D0.7778
controller.
An initial tuning value of the controller gain, Kc , can be found by solving (16) at ω = ωcp to meet the desired gain margin of
the controlled system:
Kc =
gm
gp
{(a0 − a2)2
+ a2
1}
(a0 − a2)2(1 − Ti)2 + a2
1(1 + Ti)2
; gp ̸= ∞, (29)
where gp =


Gp

jωcp
ωcp


 and gm is the desired gain margin.
If gp is undefined, one may replace it by a large value, gp = M, and start tuning Kc to meet the design requirements.
The design algorithm of the PIµ
Dµ
can be summarized as follows.
(i) Use (17) to determine the required fractional order µ for a desired phase margin ∅m.
(ii) Determine the number of modules that need to be cascaded using (18) or (19).
(iii) Determine the required controller phase angle ϕc of the controller from (28).
(iv) Calculate the integral time constant Ti from (27).
(v) Calculate the controller gain Kc using (29).
(vi) Tune Kc and Ti to meet the design requirements.
Example 2. Consider the same system discussed in Example 1. Suppose it is required to design a PIµ
Dµ
controller to stabilize
the open-loop system, and to obtain a phase margin ∅m = 45° and a gain margin of gm ≥ 20 dB.
The phase angle of the open-loop system is θp = −205.4°. Using (17), (26) and (27) gives a biquadratic module of order
µ = 0.7778 and an angle ϕc = 70°. Assume that k1 = k2 = 1; then Eqs. (9) and (10) yields a0 = 3.3781, a1 = 4.4431,
and a2 = 0.2669. Since µ < 1, from (18), a biquadratic controller with a single stage will be used. Solving (27) gives
Ti = 4.0662. Since gp = 0, one may replace gm/gp in (29) by a nonzero value, say gm/gp = 1. Solving (29) then gives a
controller gain Kc = 1.4994 × 10−4
. Substituting into (24) and testing the phase margin of the controlled system gives
a phase margin of ∅m = 3° which is less than 40°, and that does not meet the design requirements. Now, adjust Ti and
choose a new Ti = 6 × 4.0662. This gives a gain Kc = 16 × 10−4
, which improves the phase margin to ∅m = 42.2° < 45°
at ωc = 0.308 rad/s. Increasing Ti slightly to Ti = 10 × 4.0662 gives Kc = 5.946 × 10−4
, which yields a phase margin
∅m = 46.2° > 45° at ωc = 0.303 rad/s, and a gain margin gm = 31.3 dB at ωc = 2.66 rad/s. Thus, we can meet the design
requirements by using a PI0.7778
D0.7778
controller that has the following transfer function:
Gc (s) =
11.26s4
+ 30.29s3
+ 22.7s2
+ 3.132s + 0.1204
0.9015s4 + 16.19s3 + 31.22s2 + 16.19s + 0.9015
.
Fig. 4 shows the frequency response of both the controlled and the uncontrolled systems. Clearly, the gain crossover
frequency of the controlled system has migrated slightly from ωcg = 0.475 rad/s to ωcg = 0.303 rad/s due to tuning
the gain and the integration time constant.

5. Conclusions
A new design method for a PIλ
Dµ
controller is introduced that started by introducing a new structure of a proportional–
derivative (PDµ
) controller of fractional order. Since the proposed PDµ
controller exhibits a flat phase band-limited lead
contribution, it can be cascaded with similar modules to improve the robustness of the controlled system. In designing both
PDµ
and PIλ
Dµ
controllers, biquadratic approximations of fractional-order differential (or integral) operators were assumed
to be equal; i.e., λ = µ. This simplified the tuning process of the PIλ
Dµ
controller by reducing the five parameters that must
be tuned to only three: the controller gain, Kc , the integration time constant, Ti, and the order of the biquadratic module, µ.
When using a PDµ
controller, the crossover frequency of the controlled system coincides with that of the open-loop system.
However, this may not be guaranteed in the case of a PIµ
Dµ
controller due to tuning the controller gain. All the main points
of this work were verified using numerical simulation.
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